nth Roots and Rational ExponentsSolve Radical Equations

Objectives:

1. To simplify expressions involving th roots and rational exponents

2. To solve really cool (radical) equations

Objective 1You will be able to simplify expressions involving th roots and rational exponents

𝐸𝑧=𝑘𝑞𝑧

(𝑧 2+𝑅2 )3/2

Electric Field due to a Ring of Charge

Square Roots and Beyond

The number is a square root of if .• This is usually written

Radicand

Radical

Square Roots and Beyond

The number is a square root of if .• This is usually written

Any positive number has two

real square roots, one positive and

one negative, and

and , since 22 = 4 and (−2)2 = 4

The positive square root is considered the principal square root

Square Roots and Beyond

Likewise, is the cube root of if .• This is usually written

Index

Square Roots and Beyond

Likewise, is the cube root of if .• This is usually written

Any positive or negative number has one real cube root, and the other two are imaginary

, since

, since

Square Roots and Beyond

Finally, is the th root of if .• This is usually written

Index

On a your calculator, cube and th roots can be found in the MATH

Menu

Exercise 1

Use a calculator to evaluate the following th roots.

1.

2.

3.

4.

5.

6.

7.

8.

Rational Exponents

Square, cube, nth roots can be written using rational exponents. In other words, roots have fractional exponents.

(√𝑎)2=𝑎

(𝑎𝑥 )2=𝑎

Let𝑎𝑥=√𝑎So𝑎1/2=√𝑎

𝑎2𝑥=𝑎𝑎2𝑥=𝑎1 2 𝑥=1

𝑥=12

Rational Exponents

Square, cube, nth roots can be written using rational exponents. In other words, roots have fractional exponents.

(𝑏𝑥 )3=𝑏𝑏3𝑥=𝑏𝑏3𝑥=𝑏1 3 𝑥=1

𝑥=13

( 3√𝑏)3=𝑏 Let𝑏𝑥= 3√𝑏So𝑏1 /3=3√𝑏

Rational Exponents

Square, cube, nth roots can be written using rational exponents. In other words, roots have fractional exponents.

(𝑐 𝑥)𝑛=𝑐

𝑐𝑛𝑥=𝑐𝑐𝑛𝑥=𝑐1 𝑛𝑥=1

𝑥=1𝑛

(𝑛√𝑐 )𝑛=𝑐 Let 𝑐𝑥=𝑛√𝑐So𝑐1/𝑛=𝑛√𝑐

Exercise 2

Without a calculator to evaluate the following.

1. 2. 3. 4.

Objective 1You will be able to simplify expressions involving th roots and rational exponents

𝐸𝑧=𝑘𝑞𝑧

(𝑧 2+𝑅2 )3/2

Electric Field due to a Ring of Charge

Real nth Roots

In other words, even roots have two solutions, a positive and negative, and the

radicands have to be nonnegative.

Real nth Roots

Furthermore, odd roots only have one solution, with the same sign as the radicand,

which can be positive or negative.

Real nth Roots

Odd roots can only have one real solution, all others are imaginary.

More Rational Exponents

Exponents can be any rational number: a positive or a negative, a proper or an improper fraction.

Let be an th root of , and let be a positive integer.

Exercise 3

Without a calculator, evaluate the following.

1. 2.

1. 2.

Exercise 4

Use a calculator to approximate the following.

Solving Equations

Recall the inverse of squaring a number is taking the square root.

Similarly, the inverse of raising a number to the nth power is taking the

nth root.

We can use this relationship to solve certain equations involving th powers.

Exercise 5

Solve each equation.

1. 2.

Objective 2You will be able to solve really cool (radical) equations

Really Cool Equations

The equations below are all examples of radical equations.

The radicals involved can be of any index or can even use rational

exponents.

√5 𝑥+1=63√𝑥−10=−3

𝑥−6=√3𝑥√ 4 𝑥+1=√𝑥+10

Really Cool Equations

The equations below are all examples of radical equations.

To solve these awesome equations, you first have to

isolate the radical expression, and then raise both sides of the equation

to some power to make the radical mathemagically

disappear.

√5 𝑥+1=63√𝑥−10=−3

𝑥−6=√3𝑥√ 4 𝑥+1=√𝑥+10

Exercise 6

Solve the equation. Check your solution.3√𝑥+10=37

Step 3

Step 2

Step 1

Solving Radical Equations

To solve radical equations:

Isolate the

radical Raise each side to some power

Solve new polynomial equation

Square or cube both sides…

Exercise 7

Solve the equation. Check your solution.

√5 𝑥−9=11

Exercise 8

Solve the equation. Check your solution.

Exercise 9

Solve the equation. Check your solution.

√𝑥−6=𝑥−8

Extraneous Solutions

As the previous exercise demonstrated, it is important to check your solutions because at least one of them may be extraneous. This means that it is an apparent solution that doesn’t actually work in the original equation.

Before squaring:

−1=1

After squaring:

(−1 )2=12

1=1

Exercise 10

Solve the equation. Check your solution.

√10 𝑥+9=𝑥+3

Objective 2You will be able to solve really cool (radical) equations

Step 1

Step 3 Step 2

Step 1

Squaring Ad Nauseum

Really radical equations contain more than one radical expression. To solve these equations:

Separate radicals

Square both sides

Isolate radical

Solve and

check

Exercise 11

Solve the equation. Check your solution.

√𝑥+6=√11−𝑥−3

Exercise 12

Solve the equation. Check your solution.

√𝑥+6−2=√𝑥−2

Rational Exponents

Solving equations involving rational exponents is similar to solving radical equations.

1. Isolate the variable/expression with rational exponents

2. Raise both sides to the reciprocal power

3. Solve and check your answer(s)

(𝑥−4 )2/3−9=16

Exercise 13

Solve the equation. Check your solution.

(𝑥−4 )2/3−9=16

Exercise 14

Solve the equation. Check your solution.

(𝑥+2 )2/3+3=7

Substitution

Sometimes, using a clever substitution for an expression with a rational exponent can simplify solving an equation. For example, for the equation

we can let . Then the equation becomes(3 𝑥+1 )2/3+3 (3 𝑥+1 )1/3+2=0

𝑘2+3𝑘+2=0

Exercise 15

Solve the equation. Check your solution.(3 𝑥+1 )2/3+3 (3 𝑥+1 )1/3+2=0

6.1: nth Roots and Rational Exponents6.6: Solve Radical Equations

Objectives:

1. To simplify expressions involving th roots and rational exponents

2. To solve really cool (radical) equations